Experimental Investigation on the Cavitation Fundamental Characteristics of a Venturi Tube Under Ambient-Pressure Conditions

Abstract

The sensitivities of the cavitation inception conditions and the cavitation discharge coefficient, together with the path independence and uniqueness of inception, underpin reliable use of cavitating Venturi tube under ambient conditions. With a downstream sweep rate of 1/6 °/s verified in preliminary tests, forward and backward quasi-dynamic cavitation evolution processes were sampled to construct pressure- and flow-rate-control paths for the Venturi tube used in this study. Analysis of the control paths shows that, as the critical inlet pressure approaches infinity, the critical pressure ratio and the cavitation discharge coefficient approach characteristic values of 0.6364 and 0.5573. With ±3% practical equivalence margins, they become insensitive to the critical inlet pressure above thresholds of 2.4496 and 1.7167 MPa, respectively. The inception conditions are unique; hysteresis in the critical pressure ratio emerges for critical inlet pressures of 0.40–0.45 MPa, and path independence no longer holds for critical inlet pressure exceeding 0.45 MPa. The findings indicate that detecting and controlling Venturi cavitation under ambient conditions must account for the sensitivities and path-dependence of cavitation characteristics, and they provide a useful baseline reference.

1. Introduction

The Venturi tube is a widely used flow measurement and cavitation generation device. Based on Bernoulli’s principle, it consists of a converging section, a throat section, and a diverging section. As the fluid accelerates through the constricted throat, its velocity increases while static pressure decreases, thereby creating a measurable pressure differential which is used to determine the flow rate accurately [1]. Due to its low flow resistance and stable discharge coefficient under non-cavitating conditions, the Venturi tube has been standardized in multiple countries as a primary flow measurement device [2,3,4,5,6,7,8]. International and national standards such as ISO 5167-4 [2], GB/T 2026.4 [3], and ASME MFC-3M [4] specify geometric dimensions, material requirements, manufacturing tolerances, and provide experimentally validated discharge coefficients for non-cavitating flow within defined Reynolds number ranges.

Beyond flow measurement, Venturi tubes can generate intense cavitation—the formation of vapor bubbles when local pressure drops below the fluid’s vapor pressure [9,10]. Under cavitating conditions, Venturi tubes can maintain stable flow rates that are largely independent of downstream pressure variations, making them valuable as cavitation generators and flow control elements in wastewater treatment, heavy oil viscosity reduction, aerospace systems, and nuclear power safety applications [11,12,13,14,15].

Cavitation intensity can be characterized by the cavitation number, which has multiple formulations [9,10]. By measuring the velocity and pressure at reference locations when cavitation occurs, the velocity-based critical cavitation number can be determined; similarly, by measuring the upstream and downstream pressures of the cavitation generator, the pressure-based critical cavitation number can be obtained [16]. For Venturi tubes, many researchers have adopted the downstream-to-upstream pressure ratio as an equivalent metric [11,12,13,15,17], which shows a one-to-one correspondence with the cavitation number [11,13].

The flow rate through a Venturi tube is calculated using the discharge coefficient, defined as the ratio of actual to theoretical flow rate [1]. This coefficient accounts for viscous losses and flow separation effects and is proportional to the square root of the pressure difference. Under non-cavitating conditions, the discharge coefficient exhibits a definite relationship with Reynolds number within certain ranges [2]. However, cavitation fundamentally alters this relationship.

For Venturi tubes embedded in pipeline systems—where a cylindrical pipe matching the Venturi outlet diameter is connected downstream (Figure 1a)—extensive research has characterized cavitation behavior. Abdulaziz [11] experimentally investigated a small Venturi tube and found that cavitation occurs at the same cavitation number regardless of upstream pressure. Once cavitation is established, the discharge coefficient remains constant, independent of the cavitation number. Long et al. [13] demonstrated through experiments at various inlet pressures that cavitation occurs at identical critical pressure ratios and cavitation numbers regardless of inlet pressure. After cavitation inception, the flow rate remains nearly constant, independent of outlet pressure. Ghassemi and Fasih [12] showed that when the outlet-to-inlet pressure ratio drops below 0.8, the mass flow rate becomes constant and independent of downstream pressure. These studies collectively establish that for pipeline-embedded Venturi tubes, cavitation inception occurs at a fixed critical pressure ratio independent of inlet pressure [8,9,10], and the cavitation discharge coefficient is constant for a given geometry [8,9,10].

This behavior demonstrates three key characteristics for pipeline-embedded systems: (1) Sensitivity pattern—cavitation inception depends solely on the pressure ratio, not on inlet pressure independently; (2) Path independence—inception depends only on the current pressure ratio, not on how that ratio was reached; (3) Uniqueness—for any given pressure ratio, cavitation either consistently occurs or consistently does not occur. These characteristics enable the pressure ratio to be used as a reliable control parameter and facilitate robust flow rate control.

However, many practical applications operate under different boundary conditions. In processes such as cavitation-assisted heavy oil viscosity reduction [18], micro/nanobubble generation [19], and certain wastewater treatment applications [20], Venturi tubes discharge directly into ambient pressure environments (Figure 1b). The outlet flow expands into a space with characteristic dimensions much larger than the nozzle outlet, complicating the measurement of outlet pressure. In such configurations, ambient pressure, which is often easier to measure and typically a known operating condition, leads practitioners to use the ambient-to-inlet pressure ratio as the control parameter.

Unlike a pipeline-embedded system where extensive research has established fundamental characteristics, the behavior of Venturi tube under ambient-pressure conditions remains poorly understood. Under ambient-pressure operation, cavitation may occur at different inlet pressures, in which case the ambient-to-inlet pressure ratio is not constant. The cavitation inception conditions are therefore determined by both the critical pressure ratio and the critical inlet pressure, and the critical pressure ratio can be sensitive to the critical inlet pressure. Similarly, the cavitation discharge coefficient may vary with the critical inlet pressure. In addition, it is unknown whether cavitation inception conditions exhibit path independence and uniqueness. These knowledge gaps have practical implications:

If cavitation inception is not unique under ambient conditions, identical control protocols may yield inconsistent results, compromising the reliability of threshold setting, alarm systems, and repeatable testing. If inception is path-dependent, calibration data obtained under one pressure-adjustment scheme cannot reliably inform operation under different schemes, even for the same geometry and fluid properties.

If the critical ambient-to-inlet pressure ratio varies with inlet pressure, the presence of cavitation depends on both variables rather than the ratio alone, necessitating more complex monitoring and more conservative control strategies. Similarly, if the cavitation discharge coefficient varies with inlet pressure, stable flow prediction and adjustment cannot rely on a single characteristic coefficient; instead, both inlet pressure and the coefficient must be jointly evaluated and controlled.

This study experimentally investigates a Venturi tube operating under ambient pressure, examining the sensitivities of the critical pressure ratio and the cavitation discharge coefficient to the critical inlet pressure, as well as the uniqueness and path independence of cavitation inception conditions. These characteristics directly inform detection and control of cavitation inception and the prediction and regulation of post-inception flow in industrial practice.

2. Research Questions and Planned Decision Rules

2.1. About Sensitivities of the Cavitation Inception and Cavitation Discharge Coefficient

• Research question

Whether the critical pressure ratio and the cavitation discharge coefficient remain constant with respect to the critical inlet pressure.

• Decision Rule

From multiple independent replicates under different cavitation evolution processes, compute the mean critical pressure ratio, mean critical inlet pressure, and mean cavitation discharge coefficient as point estimates, and evaluate their Type A uncertainties. Fit functional relationships of the critical pressure ratio and the cavitation discharge coefficient versus the critical inlet pressure using the point estimates and their uncertainties and evaluate the uncertainties of function parameters via uncertainty propagation. If there exists a sufficiently wide interval of the critical inlet pressure over which the fitted value of the critical pressure ratio or the cavitation discharge coefficient remains within the prespecified practical equivalence margins relative to a characteristic level, then that quantity is deemed constant over this interval and thus insensitive to the critical inlet pressure. The uncertainty of the interval boundaries is obtained by propagating the uncertainties of the fitted function parameters.

2.2. About Uniqueness of Cavitation Inception Conditions

• Research question

Whether repeated cavitation evolution processes yield consistent critical pressure ratio and critical inlet pressure.

• Decision Rule

If the maximum difference among independent replicates in the critical pressure ratio and the maximum difference in the critical inlet pressure both lie within the prespecified equivalence margins, then the repeated cavitation evolution processes are considered to have consistent critical pressure ratio and critical inlet pressure, and the cavitation inception conditions are unique.

2.3. About Path Independence of Cavitation Inception Conditions

• Research question

Whether forward and backward cavitation evolution processes yield consistent critical pressure ratio and critical inlet pressure.

• Decision Rule

Using multiple independent replicates for both forward and backward processes, compute the mean critical pressure ratio and mean critical inlet pressure (point estimates) and their Type A uncertainties. Evaluate the differences between the forward and backward point estimates and the 90% confidence intervals of these differences for comparison against the two-sided prespecified equivalence margins. Use the point-estimate difference as the primary criterion and the 90% confidence interval as supporting evidence. If the forward–backward differences in both the mean critical pressure ratio and the mean critical inlet pressure lie within the prespecified equivalence margins, then the forward and backward processes are considered consistent, and the cavitation inception conditions are path independent.

3. Experimental System and Methods

3.1. Dual-Valve Cavitation System

Figure 2 illustrates the cavitation measurement and control system, while

Table 1. Specifications of the main components in the dual-valve cavitation system.

Component Name	Specifications
Filter	Filtration precision: 500 mesh (25 μm)
Centrifugal Pump	Rated/Max flow rate: 25/30 m3/h Rated/Max head: 20/48 m
Ball Valves	DN25 PN16; Opening range: 0–90°
Pressure Transmitters (HPM180H, HIGHJOIN Ltd., Nanjing, China)	Accuracy: ±0.25% of full scale Range: 0–0.6 MPa Maximum sampling rate: 512 Hz
Turbine Flowmeter (LWGY-YBLHD25, Youniu Ltd., Shanghai, China)	Accuracy: ±0.5% of full scale Range: 0.5–12 m3/h Maximum sampling rate: 4 Hz

lists the main specifications of its key components.

In the experimental system, the turbine flowmeter is installed between the nozzle and the filter to avoid interference from cavitation-induced gas bubbles generated by the Venturi tube and ball valves [21]. A straight pipe with a length of 50 times the inlet diameter of the flowmeter is placed upstream to ensure a fully developed, swirl-free flow at the flowmeter inlet, thereby ensuring the accuracy of the flow rate measurements [22]. To ensure a fully developed turbulent flow entering the Venturi tube, a straight pipe with a length of 40 times the inlet diameter of the Venturi tube is installed upstream [23]. As illustrated in Figure 2b, the ambient-pressure chamber, made of polymethyl-methacrylate (PMMA, also known as Perspex), has a diameter of 130 mm and a length of 1000 mm, which are 10.4 and 80 times the Venturi tube’s inlet diameter, respectively. To prevent gas accumulated at the top from affecting the jet flow after cavitation occurs, the chamber is slightly inclined, allowing gas to be vented through a gas-release valve. The gas-release valve (Pq21x-10p, Ruizide Ltd., Tanshan, China) allows gas to be vented while maintaining a stable ambient pressure. To minimize interference from the jet flow, downstream contraction, and gas-phase motion at the top of the ambient-pressure chamber on the ambient-pressure measurements, the ambient-pressure tap is placed at the center of the bottom side of the chamber.

The dimensioned schematic of the nylon Venturi tube used in this study is shown in Figure 3. Following previous studies [23,24], the convergent half-angle α_con is 18°, and the divergent half-angle α_div is 9°. The throat diameter d_th, inlet diameter d_i and outlet diameter d_o are 5, 12.5 and 14.84 mm, respectively, giving a diameter ratio β of 0.40. The convergent and divergent sections meet directly without a cylindrical throat; thus, the throat length L_th is zero and the throat-to-diameter ratio L_th/d_th is zero.

Since the Venturi tube and ambient-pressure chamber are located between the upstream and downstream valves, under non-cavitating conditions, both the ambient pressure and inlet pressure tend to decrease with decreasing upstream valve opening and increase downstream valve opening. Meanwhile, the ambient-to-inlet pressure ratio decreases, and the flow rate increases as both upstream and downstream valve openings increase. When the downstream valve is fully closed, the maximum pressure generated by the system is 0.48 MPa; as shown in Figure 3, when both upstream and downstream valves are fully open, the maximum flow rate of the system is 1.3 m³·h⁻¹.

To provide sufficient space for installing and removing the Venturi tube, the upstream pressure tap used to measure the Venturi inlet pressure is positioned 0.3 m upstream of the Venturi inlet. The tube between the upstream tap and the Venturi inlet has the same inner diameter as the Venturi inlet (12.5 mm) and an absolute roughness of approximately 3.2 μm. Considering the system’s maximum flow rate, Appendix F.2, gives a calculated maximum frictional pressure drop of 2.4 × 10⁻³ MPa. According to the analysis in Appendix F.2, the upstream tap pressure is an acceptable proxy for the Venturi inlet pressure.

Compared with the pressure transmitters, the turbine flowmeter has the lowest maximum sampling frequency, and therefore, the synchronous acquisition of pressure and flow signals is limited to a system sampling rate of 4 Hz.

3.2. Water Property Control

The working fluid is fresh tap water. Because the reservoir volume is limited and the maximum flow rate of the system exceeds the make-up flow, the water is recirculated. Cavitation can rapidly alter water quality, including bubble nuclei, dissolved gas content, and temperature. To limit the water-property drift over the duration of at least three independent replicates and thereby avoid confounding effects, fresh tap water is continuously supplied to the reservoir at the maximum feasible inflow rate while an equal outflow rate is maintained, as illustrated in Figure 2a.

To monitor water properties, the temperature and dissolved oxygen concentration are measured before each experiment. Dissolved oxygen concentration is measured using a polarographic dissolved oxygen meter, as shown in Appendix F, Figure A13.

3.3. Experimental Method

3.3.1. Pressure-Control Path and Flow-Control Path

Due to the use of a dual-valve throttling approach to regulate the upstream and downstream pressures of the Venturi tube, adjusting the downstream valve under a fixed upstream valve-opening results in simultaneous changes in the inlet pressure and the pressure ratio along a specific path, referred to as the pressure-control path in this study.

In the non-cavitating regime, increasing the downstream valve opening leads to a synchronous decrease in both the inlet pressure and the pressure ratio. Once cavitation occurs, the throat pressure of the Venturi tube reaches the saturated vapor pressure. After cavitation inception, the pressure ratio continues to decrease as the downstream valve opening increases, whereas the inlet pressure and the flow rate remain approximately at their inception values [11,12,13,14,15]. The cavitation inception conditions—defined by the critical pressure ratio and the critical inlet pressure—can be determined from the intersection of the non-cavitating and cavitating pressure-control paths.

Similarly to the pressure-control path, the synchronous variation of the pressure ratio and flow rate is referred to as flow-rate-control path. Based on the stabilized flow rate and inlet pressure in the cavitating regime, the cavitation discharge coefficient can be determined.

3.3.2. Static and Quasi-Dynamic Experiments

To obtain the pressure and flow-rate-control paths under a fixed upstream valve opening, the downstream valve can either be adjusted in discrete steps or continuously varied, from fully open to fully closed. In the case of adjusting the downstream valve in discrete steps, referred to as the static experiment in this study, the pressure and flow rate are synchronously sampled over time for each valve opening step. The resulting mean values at each setting define discrete nodes with high measurement accuracy, from which the pressure and flow-rate-control paths, referred to as static control paths in this study, are constructed.

To ensure that the difference between adjacent nodes along the path is significantly larger than the system’s measurement uncertainty, the discrete downstream valve openings should be spaced sufficiently apart in static experiments. However, this requirement leads to sparsely distributed nodes, resulting in reduced path resolution and limiting the ability to capture the details of the cavitation evolution process.

In contrast, quasi-dynamic experiments, which involve the continuous adjustment of the downstream valve, allow the pressure and flow rate to be sampled in real time. This enables the construction of dense control paths, referred to as quasi-dynamic control paths in this study, offering improved resolution for capturing the details of the cavitation evolution process. However, due to the potential influence of a rapid valve sweep rate on cavitation inception and the limited sampling bandwidth of the measurement system, the cavitation inception conditions and cavitation discharge coefficients derived from the static and quasi-dynamic control paths may differ.

3.3.3. Downstream Valve Sweep Rate and Sampling Rate

The pressure variation rate during quasi-dynamic experiments, which is controlled by the downstream valve sweep rate, influences the cavitation evolution process. An excessively fast downstream valve sweep rate can cause the pressure-control path to deviate significantly from that of the static experiment, thereby, yielding different cavitation inception conditions. In this case, the cavitation evolution becomes a dynamic process.

Therefore, an appropriate downstream valve sweep rate that ensures a quasi-dynamic cavitation evolution process should be preliminarily determined by assessing the equivalence between the cavitation inception conditions derived from the static and quasi-dynamic control paths constructed from the 512 Hz pressure records. With the downstream valve sweep rate determined and the equivalence of the cavitation inception conditions established, and, given that the flow rate data are not required for the hysteresis behavior analysis of cavitation inception, the analysis of path independence and uniqueness is conducted based on the 512 Hz pressure records from the quasi-dynamic experiments.

The sensitivity analysis of cavitation inception conditions and the cavitation discharge coefficient requires synchronized 4 Hz pressure and flow rate data. The commonly occurring 10–100 Hz pressure oscillations associated with cloud shedding in Venturi flow, as well as the pressure oscillations with higher frequencies, cannot be resolved by 4 Hz pressure records Consequently, instantaneous inception metrics identified from 4 Hz data may be biased if inception overshoot/undershoot events occur but are not captured. In the present study, an intersection-based procedure is used to determine the inception conditions. Non-cavitating inlet-pressure and pressure-ratio data are fitted by least squares to obtain the non-cavitating pressure-control path, while the cavitating segment is represented by the stable plateau, the mean of the inlet pressure. The inception condition is then determined as the intersection of these two relations.

Although cavitation inception in this study is determined using an intersection-based method, the low sampling rate is conservatively regarded as a potential source of bias between the cavitation inception conditions derived from the static and quasi-dynamic control paths with the system sampling rate of 4 Hz. Under the determined downstream valve sweep rate, the potential bias introduced by the system sampling rate may cause discrepancies between the cavitation inception conditions derived from the static and quasi-dynamic 4 Hz records, as well as between their corresponding cavitation discharge coefficients. It is necessary to verify whether, under the determined downstream valve sweep rate, the cavitation inception conditions obtained from the static and quasi-dynamic experiments at a system sampling rate of 4 Hz remain equivalent.

The equivalence between the cavitation inception conditions derived from the static and quasi-dynamic pressure-control paths measured at a sampling rate of 512 Hz, together with the equivalence between those obtained at 4 Hz and the equivalence of the cavitation discharge coefficients derived at 4 Hz, can only indicate that the cavitation inception conditions and cavitation discharge coefficients obtained from both 512 Hz and 4 Hz records characterize quasi-dynamic cavitation evolution processes. If the cavitation inception conditions derived from the 512 Hz and 4 Hz records are not equivalent, it implies that the pressure characteristics of cavitation evolution processes characterized by the 512 Hz and 4 Hz data are not identical, although the actual physical cavitation evolution process remains the same under the determined downstream valve sweep rate and identical experimental conditions. Therefore, to ensure that the cavitation inception conditions derived from the 512 Hz and 4 Hz records consistently characterize the same pressure characteristics of the quasi-dynamic cavitation evolution process, the equivalence between the cavitation inception conditions obtained from the 512 Hz and 4 Hz pressure records must be verified.

Since the flow rate data at 512 Hz are unavailable, the equivalence of the cavitation discharge coefficients derived from the 512 Hz and 4 Hz records in characterizing the flow-rate characteristics of the quasi-dynamic cavitation evolution process cannot be directly verified. Therefore, the synchronized pressure–flow records at 4 Hz are systematically downsampled to multiple effective sampling rates with various phase offsets. For each case, the cavitation discharge coefficient is calculated, and the coefficients are then averaged across the phase offsets for each effective sampling rate (Phase-swept decimation validity, procedure in Appendix B.3). The equivalence between the cavitation discharge coefficients obtained from the effective sampling rates and those derived from the original 4 Hz records is used to evaluate the sensitivity of the cavitation discharge coefficient to variations in sampling rate.

3.3.4. Practical Equivalence Margins, Uncertainty Budget and Propagation

• Practical equivalence margins

During the above-mentioned equivalence verifications, the differences in critical pressure ratios, critical inlet pressures and cavitation discharge coefficients are compared with the practical equivalence margins. For the difference in pressures, i.e., critical inlet pressure, the practical equivalence margins are set to [−2 MPE, 2 MPE], where MPE is the maximum permissible error of pressure transmitters, and MPE = 0.25% × 0.6 = 1.5 × 10⁻³ MPa. For the equivalence assessment of critical pressure ratios and cavitation discharge coefficients, the relative difference, defined as Rd = (value1 − value2)/value2 × 100%, is compared with the practical equivalence margins of [−3%, 3%]. The practical equivalence margins used in the decision rules of the research questions are ±2 MPE and ±3%, and this convention is applied uniformly throughout the study.

For the difference between means, a two-sided 90% t-confidence interval is reported as a precision and credibility aid. The interval is constructed using Welch’s standard error and the Welch–Satterthwaite degrees of freedom, with the significance level set at α = 0.05 for each bound. For the difference between a sample mean and a single-experiment value, the standard error of the single-experiment value required to construct the 90% t-confidence interval is estimated using 1000 stationary bootstrap replicates of the raw samples, with an average block length of L = n^1/3 and a restart probability of p = 1/L, where n is the number of raw samples.

2.

Uncertainty budget and propagation

For the quantities involved in the equivalence assessment of a single experiment (i.e., the critical pressure ratio and critical inlet pressure of a single static experiment), their Type A uncertainties are evaluated using 1000 stationary bootstrap replicates of the raw samples. For the means obtained from multiple independent replicates, their Type A uncertainties are estimated directly from the variance of the averaged quantities and the number of independent replicates.

For directly measured quantities (pressure and flow rate), the Type B uncertainty originates from the instrument accuracy specifications. Instrument accuracy is modeled at the measurement-channel level as an additive bias with a rectangular distribution. Accordingly, for inlet and ambient pressure, their Type B standard uncertainty ${\mathrm{u}}_{\mathrm{B}}\left(\mathrm{P}\right) = 0.25 \% \times 0.6/\sqrt{3} = 8.67{ \times 10}^{-4}$ MPa; for flow rate, its Type B standard uncertainty ${\mathit{u}}_{\mathrm{B}}\left(\mathit{Q}\right) = 0.5 \% \times 0.6/\sqrt{3} = 3.47{ \times 10}^{-2}$ m³/h.

Since the measurement channels of inlet pressure, ambient pressure and flowrate are separate, their Type B standard uncertainties are assumed to be independent (uncorrelated).

The additive biases of the pressure and flow rate measurement channels are regarded as common-mode across different experiments. Consequently, the Type B standard uncertainty of pressure and flow rate, particularly that of the critical inlet pressure, does not reduce when averaging over replicates. Therefore, the Type B standard uncertainties of the critical inlet pressure from a single experiment and of the mean critical inlet pressure obtained from multiple independent replicates are both ${\mathit{u}}_{\mathrm{B}}\left(\mathit{P}\right)$.

The Type B standard uncertainty of the critical pressure ratio from a single experiment is estimated using the law of propagation of uncertainty, based on the first-order Taylor approximation [25]:

$$u_{\mathrm{B}}(P_{\mathrm{r}}^{\mathrm{c}})\approx u_{\mathrm{B}}(P)\sqrt{{\left(\frac{\mathsf{\partial }{P}_{\mathrm{r}}^{\mathrm{c}}}{\mathsf{\partial }{b}_{P_{\mathrm{i}}}}\right)}^2+{\left(\frac{\mathsf{\partial }{P}_{\mathrm{r}}^{\mathrm{c}}}{\mathsf{\partial }{b}_{P_{\mathrm{a}}}}\right)}^2}$$

(1)

$${\displaystyle \frac{\mathsf{\partial }{P}_{\mathrm{r}}^{\mathrm{c}}}{\mathsf{\partial }{b}_{P_{\mathrm{i}}}}}\approx {\displaystyle \frac{P_{\mathrm{r}}^{\mathrm{c},+P_{\mathrm{i}}}-P_{\mathrm{r}}^{\mathrm{c},-P_{\mathrm{i}}}}{2u_{\mathrm{B}}(P)}}$$

(2)

$${\displaystyle \frac{\mathsf{\partial }{P}_{\mathrm{r}}^{\mathrm{c}}}{\mathsf{\partial }{b}_{P_{\mathrm{a}}}}}\approx {\displaystyle \frac{P_{\mathrm{r}}^{\mathrm{c},+P_{\mathrm{a}}}-P_{\mathrm{r}}^{\mathrm{c},-P_{\mathrm{a}}}}{2u_{\mathrm{B}}(P)}}$$

(3)

where ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}$ is critical pressure ratio, dimensionless; ${\mathit{u}}_{\mathrm{B}}\left({\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}\right)$ is the Type B standard uncertainty of the critical pressure ratio, dimensionless; $\partial {\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}/\partial {\mathit{b}}_{\mathit{P}\mathrm{i}}$ and $\partial {\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}/\partial {\mathit{b}}_{\mathit{P}\mathrm{a}}$ are the sensitivity coefficients, dimensionless; ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c},+\mathit{P}\mathrm{i}}$, ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c},-\mathit{P}\mathrm{i}}$, ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c},+\mathit{P}\mathrm{a}}$, and ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c},-\mathit{P}\mathrm{a}}$ are bias-perturbed estimates of critical pressure ratio, dimensionless. The plus-perturbed critical pressure ratio with respect to the inlet pressure, denoted as ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c},+\mathit{P}\mathrm{i}}$ is obtained from the plus-perturbed inlet pressure samples ${\mathit{P}}_{\mathrm{i}}^{+}$ = P_i + u_B(P) and the raw ambient-pressure samples P_a, where P_i denotes the raw inlet-pressure samples. The minus-perturbed critical pressure ratio with respect to the inlet pressure, denoted as ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c},-\mathit{P}\mathrm{i}}$ is obtained from the minus-perturbed inlet pressure samples ${\mathit{P}}_{\mathrm{i}}^{-}$ = P_i − u_B(P) and P_a. The plus-perturbed critical pressure ratio with respect to the ambient pressure, denoted as ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c},+\mathit{P}\mathrm{a}}$, is obtained from the plus-perturbed ambient-pressure samples ${\mathit{P}}_{\mathrm{a}}^{+}$ = P_a + u_B(P) and P_i. The minus-perturbed critical pressure ratio with respect to the ambient pressure, denoted as ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c},-\mathit{P}\mathrm{a}}$, is obtained from the minus-perturbed ambient-pressure samples ${\mathit{P}}_{\mathrm{a}}^{-}$ = P_a − u_B(P) and P_i.

Since the pressure ratio is defined as P_r = P_a/P_i, where P_a is ambient pressure, MPa and P_i is inlet pressure, MPa. The Type B standard uncertainty of the mean critical pressure ratio obtained from multiple independent replicates is estimated using the law of propagation of uncertainty and first-order linearization of P_r = P_a/P_i as follows:

$$u_{\mathrm{B}}(\overline{P_{\mathrm{r}}^{\mathrm{c}}})\approx u_{\mathrm{B}}(P)\sqrt{{\left({\displaystyle \frac{1}{\overline{P_{\mathrm{i}}^{\mathrm{c}}}}}\right)}^2+{\left({\displaystyle \frac{\overline{P_{\mathrm{r}}^{\mathrm{c}}}}{\overline{P_{\mathrm{i}}^{\mathrm{c}}}}}\right)}^2}$$

(4)

where $\overline{{\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}}$ is the mean critical pressure ratio, dimensionless; $\overline{{\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}}$ is the mean critical inlet pressure, MPa.

Under the assumption of quasi-one-dimensional flow and saturated vapor pressure at the throat during cavitation, the stable flow rate after cavitation inception is calculated from Bernoulli’s equation and the continuity equation using the cavitation discharge coefficient, as follows:

$$Q^{\mathrm{c}}=C_{\mathrm{d}}^{\mathrm{c}}\mathrm{m}\sqrt{P_{\mathrm{i}}-P_{\mathrm{v}}}$$

(5)

where Q^c is the stable flow rate after cavitation inception, m³/h; ${\mathit{C}}_{\mathrm{d}}^{\mathrm{c}}$ is the cavitation discharge coefficient, dimensionless; P_v is the saturated vapor pressure, MPa; m is the flow capacity constant of the Venturi tube in this study, with a value of $\mathrm{m} = 3.6{ \times 10}^6\sqrt{2/\mathit{\rho }}{\left({\mathit{A}}_{\mathrm{th}}^{-2}-{\mathit{A}}_{\mathrm{i}}^{-2}\right)}^{-0.5}=3.2024$ m³ h⁻¹ MPa^−0.5 determined in this study; the throat area Ath = 1.96 × 10⁻⁵ m²; the inlet area Ai = 1.23 × 10⁻⁴ m²; ρ is the water density, kg/m³.

The Type B standard uncertainty of the mean cavitation discharge coefficient obtained from multiple independent replicates is estimated using the law of propagation of uncertainty and first-order linearization, as follows:

$$u_{\mathrm{B}}(\overline{C_{\mathrm{d}}^{\mathrm{c}}})=\overline{C_{\mathrm{d}}^{\mathrm{c}}}\sqrt{{\left({\displaystyle \frac{u_{\mathrm{B}}\left(Q\right)}{\overline{Q^{\mathrm{c}}}}}\right)}^2+{\left({\displaystyle \frac{u_{\mathrm{B}}(P)}{2\left(\overline{P_{\mathrm{i}}^{\mathrm{c}}}-P_{\mathrm{v}}\right)}}\right)}^2}$$

(6)

where $\overline{{\mathit{C}}_{\mathrm{d}}^{\mathrm{c}}}$ is the mean cavitation discharge coefficient, dimensionless; $\overline{{\mathit{Q}}^{\mathrm{c}}}$ is the mean stable flow rate, m³/h.

3.4. Experimental Sets

Experiment Set A, with its experimental conditions summarized in

Table 2. Experiment Set A.

Ku(a) (°)	Static			Quasi-Dynamic		DVSR (d)	Sampling Rate
	Exp. ID (b)	Kd (°) (a,c)	Sampling Time (s)	Exp. ID	Kd (°)
20	SF6-A20	FDS (e)	10	QF6-A20 1/2/3 (g)	10 → 55	1/6 °/s	512 Hz
	SB6-A20	BDS (f)		QB6-A20 1/2/3	55 → 10
35	SF6-A35	FDS		QF6-A35 1/2/3	10 → 55	1/6 °/s
	SF12-A35			QF12-A35		1/3 °/s
	SF18-A35			QF18-A35		1/2 °/s
	SF36-A35			QF36-A35		1 °/s
	SF216-A35			QF216-A35		6 °/s
	SB6-A35	BDS		QB6-A35 1/2/3	55 → 10	1/6 °/s
	SB12-A35			QB12-A35		1/3 °/s
	SB18-A35			QB18-A35		1/2 °/s
	SB36-A35			QB36-A35		1 °/s
	SB216-A35			QB216-A35		6 °/s
75	SF6-A75	FDS		QF6-A75 1/2/3	10 → 55	1/6 °/s
	SB6-A75	BDS		QB6-A75 1/2/3	55 → 10

^(a) K_u and K_d are the upstream and downstream ball valve openings, respectively; ^(b) The column Exp. ID refers to the Experiment Identifier, where the first letter denotes the experiment type (S for static and Q for quasi-dynamic); the second letter indicates the downstream valve adjustment direction (F for forward/increasing and B for backward/decreasing); the number following the second letter denotes the servo motor speed, which determines the downstream valve sweep rate (6 rpm for 1/6 °/s, 12 rpm for 1/3 °/s, etc.); and the final letter–number combination specifies the experiment set and the upstream valve opening; ^(c) In the static experiments, the downstream valve is adjusted while the pump remains in operation; ^(d) DVSR stands for the downstream valve sweep rate. In static experiments, the downstream valve is also adjusted according to the specified downstream valve sweep rate; ^(e) FDS refers to the Forward Discrete Sequence of downstream valve opening: 10, 15, 20, 25, 30, 35, 45 and 75°; ^(f) BDS refers to the Backward Discrete Sequence of downstream valve opening: 75, 45, 35, 30, 25, 20, 15 and 10°; ^(g) The superscripts 1, 2, and 3 indicate the three independent replicates.

, was designed as a preliminary study to determine an appropriate downstream valve sweep rate that ensures a quasi-dynamic cavitation evolution process. Experimental Set A measured the inlet and ambient pressures at a sampling rate of 512 Hz, which is significantly higher than the typical frequency range of cavitation cloud shedding (10–100 Hz) in Venturi flows. The experiments were performed at three upstream valve openings: a representative setting of 35° and the range limits of 20° and 75°. Each static experiment consists of eight tests at downstream openings of 10, 15, 20, 25, 30, 35, 45, and 75°. Each quasi-dynamic experiment with a downstream sweep rate of 1/6 °/s was conducted in three independent replicates, whereas the remaining sweep-rate conditions were executed once.

At the representative upstream valve opening of 35°, static and quasi-dynamic experiments were conducted under downstream valve sweep rates of 1/6, 1/3, 1/2, 1 and 6 °/s. Additional experiments at the upstream valve opening limits of 20° and 75° were performed to verify the validity of the identified sweep rates across the entire allowable upstream valve opening range. Considering the subsequent hysteresis analysis, Experiment Set A also included paired forward and backward-path experiments that shared the same upstream valve openings and downstream valve sweep rates. In the forward-path experiments, the downstream valve opening was adjusted from the lower limit of 10° to the upper limit of 55°, and the pressure ratio decreased; whereas in the backward-path experiments, the adjustment was made in the reverse direction, and the pressure ratio increased.

To maintain consistent conditions, all tests at a given upstream valve opening were scheduled within a single day; however, owing to the experimental workload, experiments at different openings were allowed to be conducted on separate dates.

The pressure-control paths and the corresponding cavitation inception conditions from Experiment Set A are provided in Appendix A and Appendix E. Based on the analysis in Appendix A, at a sampling rate of 512 Hz, a downstream valve sweep rate of 1/6 °/s yields equivalent cavitation inception conditions derived from the static and quasi-dynamic paths in both forward and backward experiments.

Experiment Set B, with its experiment conditions summarized in

Table 3. Experiment Set B.

Ku (°)	Static			Quasi-Dynamic		DVSR (b)	Sampling Rate
	Exp. ID	Kd (°) (a)	Sampling Time (s)	Exp. ID	Kd (°)
20	SF6-B20	FDS	60	QF6-B20 1/2/3 (c)	10 → 55	1/6 °/s	4 Hz
25				QF6-B25 1/2/3
30				QF6-B30 1/2/3
35	SF6-B35			QF6-B35 1/2/3
40				QF6-B40 1/2/3
45				QF6-B45 1/2/3
75	SF6-B75			QF6-B75 1/2/3

^(a) In the static experiments, the downstream valve is adjusted while the pump remains in operation; ^(b) In static experiments, the downstream valve is adjusted according to the specified downstream valve sweep rate. ^(c) The superscript ^1/2/3 label the three independent replicates of the experiment.

, is designed to investigates the sensitivities of the cavitation inception and the cavitation discharge coefficient with decreasing pressure ratio. Due to the flowmeter sampling constraint, the system sampling rate for synchronous pressure and flow acquisition is 4 Hz. To obtain as much data as possible for the sensitivity analysis, the downstream valve sweep rate was preliminarily determined as 1/6 °/s. Experiment Set B includes seven quasi-dynamic experiments at upstream valve openings of 20°, 25°, 30°, 35°, 40°, 45° and 75°, and three static experiments at a representative setting of 35° and the range limits of 20° and 75°. Each quasi-dynamic experiment is performed in three independent replicates. Each static experiment comprises eight tests at downstream openings of 10°, 15°, 20°, 25°, 30°, 35°, 45°, and 75°.

Static and quasi-dynamic experiments at upstream valve openings of 20°, 35°, and 75° were conducted first, at a downstream sweep rate of 1/6 °/s, to confirm the equivalence of cavitation inception conditions between static and quasi-dynamic tests at 4 Hz and to verify the equivalence of inception conditions obtained at 4 Hz and 512 Hz.

To maintain consistent conditions, all tests at a given upstream valve opening are scheduled within a single day; however, owing to the experimental workload, experiments at different openings are allowed to be conducted on separate dates.

Based on the equivalent cavitation inception conditions of the static and quasi-dynamic experiments in Experiment Set B at upstream valve openings of 20°, 35°, and 75° (presented in Appendix B.1), the cavitation evolution process is confirmed to be quasi-dynamic at the downstream valve sweep rate of 1/6 °/s.

In Appendix B.2, the equivalence of the cavitation inception conditions for the forward quasi-dynamic experiments conducted at upstream valve openings of 20°, 35°, and 75° in Experiment Sets A (512 Hz) and B (4 Hz) is presented. The equivalence of the cavitation inception conditions indicates that a system sampling rate of 4 Hz is adequate relative to the downstream valve sweep rate of 1/6 °/s for determining the cavitation inception conditions.

Appendix B.3 decimates the 4 Hz flow-rate and pressure records from experiments at upstream valve openings of 20°, 35°, and 75° in Experiment Set B to effective sampling rates of 2, 1, 0.5, 0.25, and 0.125 Hz. At each upstream valve opening, the resulting estimates of the cavitation discharge coefficient are equivalent to those computed from the original 4 Hz data. Moreover, because the cavitation discharge coefficient is calculated from the average flow rate and inlet pressure under stable cavitation conditions, its precision is governed primarily by the effective sample size rather than the nominal sampling rate. Therefore, a system sampling rate of 4 Hz is adequate for determining the cavitation discharge coefficient.

The hysteresis behavior analysis is based on the forward and backward quasi-dynamic experiments in Experiment Set A, which were conducted at the determined downstream valve sweep rate of 1/6 °/s and has a sampling rate of 512 Hz.

The hysteresis analysis is based on the forward and backward quasi-dynamic experiments in Experiment Set A, conducted at the determined downstream valve sweep rate of 1/6 °/s with a sampling rate of 512 Hz.

Before each experiment in Experiment Set A and B, the pump was stopped for 1 min to stabilize the system, and the temperature and dissolved oxygen concentration were measured and recorded.

4. Results and Discussion

4.1. Fluid Property

Figure 4 presents the temperature and dissolved oxygen concentration measurements for all experiments. The corresponding numerical data are also provided in Appendix A and Appendix C.

Owing to the continuous inflow of fresh tap water and equal outflow, the water temperature was maintained from 23.7 to 25.4 °C and the dissolved oxygen concentration ranged from 5.7 to 7.2 mg/L.

4.2. Pressure- and Flow-Rate-Control Paths

The pressure- and flow-rate-control paths for each set of experiments are illustrated in Figure 5. In the non-cavitating regime, a clear nonlinear relationship is observed between the pressure ratio and the inlet pressure, as well as between the pressure ratio and the flow rate. In the cavitating regime, both the inlet pressure and the flow rate remain constant as the pressure ratio continues to decrease, which is consistent with the findings of Ghassemi et al. [11,12,13,14,15].

Alongside the ambient-to-inlet pressure ratio, the cavitation number serves as an equivalent cavitation control variable, which can be defined as [16,26]:

$$\mathsf{\sigma }={\displaystyle \frac{P_{\mathrm{i}}-P_{\mathrm{v}}}{P_{\mathrm{i}}-P_{\mathrm{a}}}}$$

(7)

where σ is cavitation number, dimensionless. Throughout all experiments, the water temperature ranged from 23.7 to 25.4 °C, and the saturated vapor pressure of water is approximately 3.168 × 10⁻³ MPa, which is much lower than the minimum inlet pressure of approximately 0.1 MPa across experiments (see Appendix C,

Table A7. Summary of measurements for Experiment Set B.

Ku (°)	Value Type	${\mathit{P}}_{\mathbf{i}}^{\mathbf{c}}$ (MPa)	${\mathit{P}}_{\mathbf{a}}^{\mathbf{c}}$ (MPa)	${\mathit{P}}_{\mathbf{r}}^{\mathbf{c}}$ (1)	Qc (m3/h)	${\mathbf{C}}_{\mathbf{d}}^{\mathbf{c}}$ (1)	${\mathbf{v}}_{\mathbf{i}}^{\mathbf{c}}$ (m/s)	${\mathbf{v}}_{\mathbf{th}}^{\mathbf{c}}$ (m/s)	${\mathbf{Re}}_{\mathbf{th}}^{\mathbf{c}}$ (×104)	${\mathbf{Re}}_{\mathbf{i}}^{\mathbf{c}}$ (×104)	σc (1)	ΔP (×103 Pa)	OD (mg/L)	T (°C)
20	mean	0.0986	0.0164	0.1661	0.8017	0.7974	1.81	4.53	6.21	2.48	1.20	1.2
	uA	0.0004	0.0001	0.0008	0.0003	0.0013
	uB	uB(P)	uB(P)	0.0089	uB(Q)	0.0346
	1st	0.0993	0.0166	0.1674	0.8021	0.7949	1.82	4.55	6.21	2.48	1.20	1.2	5.7	24.5
	2nd	0.0984	0.0164	0.1663	0.8017	0.7980	1.81	4.53	6.21	2.48	1.20	1.2	6.3	24.0
	3rd	0.0980	0.0161	0.1647	0.8012	0.7992	1.81	4.53	6.21	2.48	1.20	1.2	6.3	24.7
25	mean	0.2611	0.1164	0.4459	1.0806	0.6604	2.45	6.13	8.37	3.35	1.80	2.1
	uA	0.0002	0.0005	0.0015	0.0005	0.0005
	uB	uB(P)	uB(P)	0.0036	uB(Q)	0.0212
	1st	0.2609	0.1162	0.4453	1.0808	0.6607	2.45	6.13	8.37	3.35	1.80	2.1	6.6	23.8
	2nd	0.2615	0.1173	0.4487	1.0797	0.6593	2.44	6.10	8.36	3.34	1.81	2.1	6.8	23.9
	3rd	0.2608	0.1157	0.4436	1.0813	0.6611	2.45	6.13	8.38	3.35	1.80	2.1	6.6	24.0
30	mean	0.3555	0.1782	0.5011	1.2070	0.6321	2.73	6.83	9.35	3.74	2.00	2.6
	uA	0.0001	0.0003	0.0007	0.0008	0.0004
	uB	uB(P)	uB(P)	0.0027	uB(Q))	0.0182
	1st	0.3553	0.1784	0.5020	1.2057	0.6317	2.73	6.83	9.34	3.74	2.01	2.6	6.7	24.1
	2nd	0.3555	0.1776	0.4997	1.2086	0.6329	2.74	6.85	9.36	3.74	2.00	2.6	6.6	24.2
	3rd	0.3558	0.1785	0.5016	1.2068	0.6318	2.73	6.83	9.35	3.74	2.01	2.6	6.5	24.3
35	mean	0.4009	0.2078	0.5184	1.2697	0.6262	2.87	7.18	9.84	3.93	2.08	2.8
	uA	0.0002	0.0007	0.0013	0.0002	0.0001
	uB	uB(P)	uB(P)	0.0024	uB(Q)	0.0171
	1st	0.4008	0.2072	0.5170	1.2698	0.6263	2.87	7.18	9.83	3.93	2.07	2.8	6.6	24.0
	2nd	0.4013	0.2091	0.5210	1.2701	0.6261	2.87	7.18	9.84	3.94	2.09	2.8	6.6	24.1
	3rd	0.4005	0.2071	0.5171	1.2693	0.6263	2.87	7.18	9.83	3.93	2.07	2.8	6.5	24.1
40	mean	0.4212	0.2251	0.5343	1.2927	0.6220	2.93	7.33	10.01	4.02	2.15	2.9
	uA	0.0003	0.0010	0.0022	0.0004	0.0002
	uB	uB(P)	uB(P)	0.0023	uB(Q)	0.0167
	1st	0.4211	0.2239	0.5316	1.2920	0.6217	2.92	7.30	10.01	4.00	2.13	2.9	6.1	24.1
	2nd	0.4208	0.2242	0.5328	1.2926	0.6223	2.93	7.33	10.01	4.00	2.14	2.9	6.1	24.3
	3rd	0.4217	0.2271	0.5386	1.2935	0.6220	2.93	7.33	10.02	4.06	2.17	2.9	6.2	24.3
45	mean	0.4347	0.2318	0.5332	1.3094	0.6201	2.96	7.40	10.14	4.07	2.14	3.0
	uA	0.0005	0.0021	0.0042	0.0007	0.0002
	uB	uB(P)	uB(P)	0.0023	uB(Q)	0.0164
	1st	0.4357	0.2359	0.5415	1.3109	0.6202	2.97	7.43	10.15	4.06	2.18		6.8	24.0
	2nd	0.4339	0.2289	0.5275	1.3086	0.6203	2.96	7.40	10.14	4.06	2.12	3.0	6.7	24.0
	3rd	0.4346	0.2306	0.5307	1.3086	0.6198	2.96	7.40	1.014	4.10	2.13	3.0	6.8	24.3
75	mean	0.4470	0.2412	0.5396	1.3248	0.6187	3.00	7.50	10.26	4.10	2.17	3.0
	uA	0.0004	0.0010	0.0022	0.0005	0.0003
	uB	uB(P)	uB(P)	0.0022	uB(Q)	0.0162
	1st	0.4467	0.2393	0.5357	1.3243	0.6187	3.00	7.50	10.26	4.10	2.15	3.0	7.8	24.2
	2nd	0.4477	0.2417	0.5398	1.3258	0.6187	3.00	7.50	10.27	4.11	2.17	3.0	7.2	24.2
	3rd	0.4466	0.2426	0.5432	1.3242	0.6188	3.00	7.50	10.26	4.10	2.19	3.0	7.2	24.4

). Therefore, by neglecting the saturated vapor pressure, the cavitation number can be expressed in terms of the pressure ratio as

$$\mathsf{\sigma }={\displaystyle \frac{1}{1-P_{\mathrm{r}}}}$$

(8)

When the cavitation number is used as the control variable, the pressure- and flow-control paths are illustrated in Figure 6.

The cavitation inception conditions, cavitation discharge coefficients and the critical cavitation numbers for Experiment Set B are provided in Appendix C, Table A7. The cavitation inception conditions, cavitation discharge coefficients and the critical cavitation numbers for Experiment Set A are provided in tables of Appendix A.

A representative group of cavitation images from forward static experiment SF6-B75 1st in Experiment Set B, which conducted at the upstream valve opening of 75°, is presented in Figure 7. The white, flocculent structures visible in Figure 7b–d represent the cavitation cloud. The cavitation occurs at a moment between Figure 7a,b.

4.3. Sensitivity of Cavitation Inception Conditions and Cavitation Discharge Coefficients

4.3.1. Sensitivity of Cavitation Inception Conditions

For each upstream valve opening in Experiment Set B, the paired means of the critical ambient pressure (i.e., the ambient pressure at cavitation inception) and the critical inlet pressure, averaged over three independent forward quasi-dynamic replicates, are presented in Figure 8.

The critical ambient pressure exhibits a clear linear relationship with the critical inlet pressure. Taking into account the Type A uncertainties of the mean critical ambient and inlet pressures, an affine model, ${\mathit{P}}_{\mathrm{a}}^{\mathrm{c}}{ = \mathrm{A}\mathit{P}}_{\mathrm{i}}^{\mathrm{c}} + \mathrm{B}$, is fitted by nonlinear orthogonal distance regression to describe the linear relationship. The accompanying 95% confidence intervals for parameters A and B are obtained via parametric Monte-Carlo simulation under a t-error model for the means.

Since the critical pressure ratio is defined as ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}{ = \mathit{P}}_{\mathrm{a}}^{\mathrm{c}}{/\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}$, the functional relationship between the critical pressure ratio and the critical inlet pressure, ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}{ = \mathrm{A} + \mathrm{B}/\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}$, is derived from the affine model of the critical ambient pressure–inlet pressure relationship, which accurately fits the mean critical ambient and inlet pressure data as shown in Figure 8.

According to the functional relationship between the critical pressure ratio and the critical inlet pressure, ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}{ = \mathrm{A} + \mathrm{B}/\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}$, as the critical inlet pressure approaches infinity, the critical pressure ratio approaches a constant characteristic pressure ratio ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c}* }= 0.6364$ MPa with a 95% t-confidence interval of [0.6301, 0.6380] MPa, obtained through parametric Monte-Carlo simulation under a t-error model for the means. Increasing the critical inlet pressure from 2.4496 MPa to infinity increases the critical pressure ratio by only 3% toward the characteristic pressure ratio. Therefore, for the Venturi tube used in this study, when the critical inlet pressure exceeds the threshold ${\mathit{P}}_{\mathrm{i}}^{\mathrm{c},\mathrm{thr}1}$ = 2.4496 MPa with a 95% t-confidence interval of [2.4172, 2.4814] MPa, the critical pressure ratio becomes largely insensitive to critical inlet pressure.

4.3.2. Sensitivity of Cavitation Discharge Coefficients

The mean cavitation discharge coefficients, the mean stable flow rates after cavitation inception and the mean critical inlet pressures are shown in Figure 9.

A root-linear model, ${\mathit{C}}_{\mathrm{d}}^{\mathrm{c}} = {\left({\mathrm{C} + \mathrm{D}/\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}\right)}^{0.5}$, is fitted to the paired means of cavitation discharge coefficient versus critical inlet pressure using nonlinear orthogonal distance regression. The 95% confidence intervals for C and D are obtained through parametric Monte-Carlo simulation under a t-error model for the means.

The stable flow rate–critical inlet pressure relationship, ${\mathit{Q}}^{\mathrm{c}}=\mathrm{k}{\left({\mathrm{C}\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}+\mathrm{D}\right)}^{0.5}$, derived from substituting the root-linear model, ${\mathit{C}}_{\mathrm{d}}^{\mathrm{c}}={\left({\mathrm{C}+\mathrm{D}/\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}\right)}^{0.5}$, into Equation (5), agrees well with the experimental mean data as shown in Figure 9.

The root-linear model describing the cavitation discharge coefficient–critical inlet pressure relationship, when mapped through the physics-based relation in Equation (5), reproduces the measured stable flow rate–critical inlet pressure data accurately, thereby confirming the rationale of the chosen functional form and parameter estimates.

According to the functional relationship, ${\mathit{C}}_{\mathrm{d}}^{\mathrm{c}}={\left({\mathrm{C}+\mathrm{D}/\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}\right)}^{0.5}$, as the critical inlet pressure approaches infinity, the critical discharge coefficient approaches a characteristic value ${\mathit{C}}_{\mathrm{d}}^{\mathrm{c}*}=0.5573$ with a 95% t-confidence interval of [0.5563, 0.5582]. Increasing the critical inlet pressure from 1.7167 MPa to infinity results in only a 3% reduction in the critical pressure ratio towards the characteristic cavitation discharge coefficient. When the critical inlet pressure exceeds the threshold ${\mathit{P}}_{\mathrm{i}}^{\mathrm{c},\mathrm{thr}2}$ = 1.7167 MPa with a 95% t-confidence interval of [1.6780, 1.7570] MPa, the cavitation discharge coefficient becomes effectively insensitive to the critical inlet pressure.

The inlet and throat velocities can be computed from the volumetric flow rate as follows:

$$v_{\mathrm{i}}={\displaystyle \frac{277.78Q}{\frac{\pi }{4}{d}_{\mathrm{i}}^2}}$$

(9)

$$v_{\mathrm{th}}={\displaystyle \frac{277.78Q}{\frac{\pi }{4}{d}_{\mathrm{th}}^2}}$$

(10)

where v_i is the mean velocity at the Venturi inlet, m/s; vth is the mean velocity at Venturi throat, m/s. The Reynolds number based on the throat diameter is:

$${\mathrm{Re}}_{\mathrm{th}}={\displaystyle \frac{v_{\mathrm{th}}{d}_{\mathrm{th}}}{\nu }}$$

(11)

where ν is the kinematic viscosity, within the temperature range of 23.7–25.4 °C during experiments, ν = 0.9131 × 10⁻⁶ m²/s. Based on the functional relationship, ${\mathit{Q}}^{\mathrm{c}} = \mathrm{k}{\left({C\mathit{P}}_{\mathrm{i}}^{\mathrm{c}} + \mathrm{D}\right)}^{0.5}$ the critical Reynold number based on Venturi throat diameter, dimensionless and the critical velocities are in a monotonic one-to-one correspondence: ${\mathrm{Re}}_{\mathrm{th}}^{\mathrm{c}} = \mathrm{K}{\left({C\mathit{P}}_{\mathrm{i}}^{\mathrm{c}} + \mathrm{D}\right)}^{0.5}$, where K is the Reynolds-number conversion factor, K = 2.4807 × 10⁵ MPa^−0.5. Therefore, for fixed Venturi geometry and fluid properties, any functional relationship expressed with the critical inlet pressure as the independent variable can be equivalently reparameterized in terms of the critical throat Reynolds number, the critical mean inlet velocity, or the critical mean throat velocity.

The top secondary x axis in Figure 8 and Figure 9 display the critical throat Reynolds number, reparameterized from the critical inlet pressure. The critical throat Reynolds numbers, critical mean inlet velocities, and critical mean throat velocities for the experiments in Experiment Set B are provided in Appendix C, Table A7.

4.4. Path Independence and Uniqueness of Cavitation Inception Conditions

4.4.1. Hysteresis Behavior

The forward and backward pressure-control paths of the quasi-dynamic experiments conducted at upstream valve openings of 20°, 35°, and 75° with a downstream valve sweep rate of 1/6 °/s, as well as those at an upstream valve opening of 35° with a downstream valve sweep rate of 6 °/s in Experiment Set A (sampling rate: 512 Hz), are illustrated in Figure 10.

The forward and backward pressure-control paths of the quasi-dynamic experiments conducted at an upstream valve opening of 35° with a downstream valve sweep rate of 6 °/s in Figure 10c exhibit a pronounced separation. Therefore, when the inlet and ambient pressure of the Venturi tube vary rapidly, the cavitation evolution process does not exhibit complete hysteresis-free behavior.

The forward and backward pressure-control paths of the quasi-dynamic experiments conducted at upstream valve openings of 20°, 35°, and 75° with a downstream valve sweep rate of 1/6 °/s exhibit a high degree of consistency.

Table 4. Differences in critical pressure ratios and critical inlet pressures between the forward and backward quasi-dynamic experiments.

Ku (°)	Direction	${\mathit{P}}_{\mathbf{r}}^{\mathbf{c}}$ (Dimensionless)		RD (a) (%)	${\mathit{P}}_{\mathbf{i}}^{\mathbf{c}}$ (MPa)		AD (b) (10−3 MPa)
20	Forward	mean	0.1661	−2.59 [−2.77, −2.40]	mean	0.0973	1.7 [1.2, 2.2]
		uA	0.0001		uA	0.0001
		uB	0.0090		uB	uB(P)
	Backward	mean	0.1618		mean	0.0990
		uA	0.0001		uA	0.0002
		uB	0.0089		uB	uB(P)
35	Forward	mean	0.5216	−0.65 [−1.77, 0.47]	mean	0.4005	−2.1 [−3.0, −1.2]
		uA	0.0012		uA	0.0003
		uB	0.0024		uB	uB(P)
	Backward	mean	0.5182		mean	0.3984
		uA	0.0022		uA	0.0003
		uB	0.0024		uB	uB(P)
75	Forward	mean	0.5334	−4.05 [−5.31, −2.79]	mean	0.4475	−0.3 [−0.9,0.3]
		uA	0.0024		uA	≈0
		uB	0.0022		uB	uB(P)
	Backward	mean	0.5118		mean	0.4472
		uA	0.0020		uA	0.0002
		uB	0.0022		uB	uB(P)

^(a) RD is the relative difference in the critical pressure ratios. ${\mathrm{RD} = (\mathit{P}}_{\mathrm{r},\mathrm{f}}^{\mathrm{c}}-{\mathit{P}}_{\mathrm{r},\mathrm{b}}^{\mathrm{c}}{)/\mathit{P}}_{\mathrm{rf}}^{\mathrm{c}} \times 100\%$, where ${\mathit{P}}_{\mathrm{r},\mathrm{f}}^{\mathrm{c}}$ and ${\mathit{P}}_{\mathrm{r},\mathrm{b}}^{\mathrm{c}}$ are the critical pressure ratios obtained from the forward and backward quasi-dynamic experiments, respectively; ^(b) AD is the absolute difference in critical inlet pressure. ${\mathrm{AD} = \mathit{P}}_{\mathrm{i},\mathrm{f}}^{\mathrm{c}}-{\mathit{P}}_{\mathrm{i},\mathrm{b}}^{\mathrm{c}}$, where ${\mathit{P}}_{\mathrm{i},\mathrm{f}}^{\mathrm{c}}$ and ${\mathit{P}}_{\mathrm{i},\mathrm{b}}^{\mathrm{c}}$ are the critical pressures from the forward and backward quasi-dynamic experiments, respectively.

quantifies the consistency between the pressure-control paths of forward and backward quasi-dynamic experiments by the differences in the mean critical pressure ratios and mean critical inlet pressures.

For the relative difference in critical pressure ratios, the practical equivalence margins are set to [−3%, 3%]; for the absolute difference in critical inlet pressure, the margins are [−3 × 10⁻³, 3 × 10⁻³] MPa. According to Table 4, considering both the critical pressure ratio and the critical inlet pressure, cavitation inception is hysteresis-free at upstream valve openings of 20° and 35°, whereas it is not hysteresis-free at 75°, with a downstream valve sweep rate of 1/6 °/s⁻¹.

In Table 4, the absolute differences in critical inlet pressure are all smaller than the corresponding practical equivalence margins and are on the order of 10⁻³ MPa, indicating negligible discrepancies. Because ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}{ = \mathit{P}}_{\mathrm{a}}^{\mathrm{c}}{/\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}$, the observed differences in the critical inlet pressure arise primarily from the critical ambient pressure. At a downstream valve sweep rate of 1/6 °/s⁻¹, cavitation inception is hysteresis-free with respect to the critical inlet pressure across the upstream valve openings examined. By contrast, for the critical pressure ratio, under the practical equivalence margins specified in this study, hysteresis emerges at an upstream valve opening between 35° and 75° as the opening increases.

4.4.2. Uniqueness of Cavitation Inception Conditions

Table 5. Maximum differences in cavitation inception conditions between single forward experiments in Experiment Set A.

Ku (°)	Exp. Type (a)	${\mathit{P}}_{\mathbf{r}}^{\mathbf{c}}$ (Dimensionless)		RD (b) (%)	${\mathit{P}}_{\mathbf{i}}^{\mathbf{c}}$ (MPa)		AD (c) (10−3 MPa)
20	St	0.1676 max		1.58		0.0991 max	2.5
	Qd	1st	0.1670		1st	0.0966
		2nd	0.1650 min		2nd	0.0986
		3rd	0.1663		3rd	0.0966 min
35	St	0.5318 max		2.37	PE	0.4005	0.9
	Qd	1st	0.5195 min		1st	0.4010 max
		2nd	0.5238		2nd	0.4005
		3rd	0.5215		3rd	0.4001 min
75	St	0.5351		1.53		0.4467 min	1.4
	Qd	1st	0.5322		1st	0.4481 max
		2nd	0.5381 max		2nd	0.4469
		3rd	0.5300 min		3rd	0.4474

^(a) St denotes static experiments and Qd denotes quasi-dynamic experiment; ^(b) RD is the relative difference in critical pressure ratios. ${\mathrm{RD} = (\mathit{P}}_{\mathrm{r}}^{\mathrm{c},\max }-{\mathit{P}}_{\mathrm{r}}^{\mathrm{c},\min }{)/\mathit{P}}_{\mathrm{r}}^{\mathrm{c},\min } \times 100\%$, where ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c},\max }$ and ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c},\min }$ are the maximum and minimum critical pressure ratios of the experiments at an upstream valve opening; ^(c) AD is the absolute difference in critical inlet pressure. ${\mathrm{AD}= \mathit{P}}_{\mathrm{i}}^{\mathrm{c},\max }-{\mathit{P}}_{\mathrm{i}}^{\mathrm{c},\min }$, where ${\mathit{P}}_{\mathrm{i}}^{\mathrm{c},\max }$ and ${\mathit{P}}_{\mathrm{i}}^{\mathrm{c},\min }$ are the maximum and minimum critical inlet pressure of the experiments at an upstream valve opening.

reports, for Experiment Set A, the maximum differences among the inception conditions across forward single-run experiments at each upstream valve opening. All differences fall within the specified practical equivalence margins.

Table 6. Maximum differences in cavitation inception conditions between single backward experiments in Experiment Set A.

Ku (°)	Exp. Type	${\mathit{P}}_{\mathbf{r}}^{\mathbf{c}}$ (Dimensionless)		RD (%)	${\mathit{P}}_{\mathbf{i}}^{\mathbf{c}}$ (MPa)		AD (10−3 MPa)
20	St	0.1601 min		1.62	0.1003 max		1.6
	Qd	1st	0.1626 max		1st	0.0991
		2nd	0.1607		2nd	0.0987 min
		3rd	0.1620		3rd	0.0993
35	St	0.5278 max		2.53	0.3979		1
	Qd	1st	0.5176		1st	0.3985
		2nd	0.5148 min		2nd	0.3978 min
		3rd	0.5223		3rd	0.3988 max
75	St	0.5156 max		1.52	0.4470 min		0.6
	Qd	1st	0.5138		1st	0.4470
		2nd	0.5079 min		2nd	0.4471
		3rd	0.5137		3rd	0.4476 max

likewise reports the maximum differences among the inception conditions across backward single-run experiments at each upstream valve opening; again, all differences lie within the specified practical equivalence margins.

Additionally, in Experiment Set B, the maximum difference in critical pressure ratio across single forward quasi-dynamic experiments at a given upstream valve opening is 2.65%, and the maximum difference in critical inlet pressure is 1.8 × 10⁻³ MPa; both values fall within the specified practical equivalence margins.

Therefore, within the specified equivalence margins for the critical pressure ratio and the critical inlet pressure, the cavitation inception conditions of a given pressure-control path are unique.

4.4.3. Path Independence of Cavitation Inception Conditions

From the functional relationship, ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}{=\mathrm{A}+\mathrm{B}/\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}$ for the forward experiments in Experiment Set B, under the downstream valve sweep rate of 1/6 °/s, an inception locus for the Venturi tube in this study is constructed in the (${\mathit{P}}_{\mathrm{i}}^{\mathrm{c}}$, ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c}}$) plane, demarcating the non-cavitating and cavitating regimes, as shown in Figure 11.

On the inception locus, point N corresponds to the mean inception conditions for a forward experiment with an upstream valve opening within the range of 35–75°. As the opening increases past this point, hysteresis in the critical pressure ratio appears. Therefore, points on inception locus segment MN are path-independent: during a quasi-dynamic cavitation process with a sufficiently slow pressure-change rate, cavitation will occur upon reaching any point on segment MN, regardless of the specific pressure-control path.

At a downstream valve sweep rate of 1/6 °/s, when the forward pressure-control path generated by the dual-valve cavitation system intersects the inception locus at an inception point to the right of point N, the corresponding backward inception point lies within the separation band and shares the same critical inlet pressure as the forward inception point. This is illustrated by the forward and backward inception points C^f and C^b for the pressure-control paths with an upstream valve opening of 75° in Figure 11.

Under the cavitation control protocol in which the ambient pressure is held constant and the inlet pressure is varied at a sufficiently slow rate, if the pressure-control path intersects the inception locus at a point on segment MN, then the critical inlet pressure and critical pressure ratio at that point serve as the inception conditions for both forward and backward paths, as illustrated by pressure-control path I with a constant ambient pressure of 0.1 MPa in Figure 11. However, for intersections occurring to the right of point N, such as point S on pressure-control path II with a constant ambient pressure of 0.25 MPa, the intersection does not necessarily represent the forward or backward inception point since the inception locus is path-dependent in this region. Rather, the corresponding forward and backward inception points are situated within the inception separation band in the vicinity of S (schematically indicated by the red circle around point S in Figure 11).

The idealized pressure-control path at constant inlet pressure shown in Figure 11 illustrates a complementary cavitation control protocol: the inlet pressure is held fixed and quasi-dynamic cavitation evolution is driven by variations in the ambient pressure at a sufficiently slow rate of pressure change. As evidenced by the forward–backward difference in the critical inlet pressure at an upstream valve opening of 75°, which indicates that hysteresis in the critical pressure ratio stems from disparities in the critical ambient pressure, this protocol may yield the maximum forward–backward difference in critical pressure ratios, thereby defining the boundary of the inception separation band. With a cavitation experimental setup implementing this cavitation control protocol, verification of the hypothesis and construction of the separation band become feasible.

5. Discussion

5.1. Influence of Water Property

With continuous replenishment and equal discharge maintaining water temperature from 23.7 to 25.4 °C and dissolved oxygen from 5.7 to 7.2 mg/L, the differences in cavitation inception conditions across the forward and backward runs at upstream valve openings of 20°, 35°, and 75° in Experiment Set A (Table 5 and Table 6) remain within the specified practical equivalence margins. Likewise, in Experiment Set B (Appendix C, Table A7), the differences among forward quasi-dynamic independent replicates in cavitation inception conditions also fall within the specified practical equivalence margins. Therefore, when temperature and dissolved oxygen are maintained within these controlled ranges, their variations do not significantly affect the cavitation inception conditions at a downstream valve sweep rate of 1/6 °/s. For the given practical equivalence margins, the water-property control method employed in this study is effective.

5.2. Sampling Rate Considerations for Cavitation Inception and Cavitation Discharge Coefficient

When cavitation inception is defined in an instantaneous sense from time series of inlet pressure and pressure ratio, namely as the first appearance of bubbles at the instantaneous inlet pressure and pressure ratio, low sampling rates may fail to capture overshoot or undershoot in these signals. This can introduce bias in both the detected timing and the magnitude of the instantaneous inception point.

In the present study, an intersection-based procedure is used to determine the inception conditions. Non-cavitating inlet-pressure and pressure-ratio data are fitted by least squares to obtain the non-cavitating pressure-control path, while the cavitating segment is represented by the stable plateau of the mean inlet pressure after cavitation inception. The inception condition is then identified as the intersection of these two relations. Under this definition, instantaneous fluctuations do not systematically shift either the fitted path or the plateau, and the sampling rate mainly affects uncertainty through the number of points contributing to the fit rather than by introducing bias.

Appendix B,

Table A6. Differences in inception conditions between the forward quasi-dynamic experiments with sampling rates of 512 and 4 Hz under a downstream valve sweep rate of 1/6 °/s.

Ku (°)	Exp. Type	${\mathit{P}}_{\mathbf{r}}^{\mathbf{c}}$ (1)		RD (%)	${\mathit{P}}_{\mathbf{i}}^{\mathbf{c}}$ (MPa)		AD (10−3 MPa)
20°	512	mean	0.1661	0 [−1.39, 1.39]	mean	0.0973	1.3 [0.2, 2.4]
		uA	0.0001		uA	0.0001
		uB	0.0090		uB	uB(P)
	4	mean	0.1661		mean	0.0986
		uA	0.0008		uA	0.0004
		uB	0.0089		uB	uB(P)
35°	512	mean	0.5216	−0.61 [−1.34, 0.11]	mean	0.4005	0.4 [−0.4, 1.2]
		uA	0.0012		uA	0.0003
		uB	0.0024		uB	uB(P)
	4	mean	0.5184		mean	0.4009
		uA	0.0013		uA	0.0002
		uB	0.0024		uB	uB(P)
75°	512	mean	0.5334	1.16 [−0.14, 2.47]	mean	0.4475	−0.5 [−1.7, −0.7]
		uA	0.0024		uA	≈0
		uB	0.0022		uB	uB(P)
	4	mean	0.5396		mean	0.4470
		uA	0.0022		uA	0.0004
		uB	0.0022		uB	uB(P)

shows that the differences between inception conditions obtained at 4 Hz and 512 Hz fall within the practical equivalence margins specified in this study, which confirms the sampling-rate insensitivity of the intersection-based estimate. Likewise, the cavitation discharge coefficient computed from the steady flow rate and steady inlet pressure during cavitation is insensitive to sampling rate and to flow transients. Appendix B, Figure A7 shows that the differences between coefficients derived from downsampled data and those from the 4 Hz case lie within the practical equivalence margins.

5.3. Applicability Across Reynolds Numbers and Venturi Geometries

Since the monotonic one-to-one relationship between the critical Reynolds number and the critical inlet pressure, ${\mathrm{Re}}_{\mathrm{th}}^{\mathrm{c}} = \mathrm{K}{\left({\mathrm{C}\mathit{P}}_{\mathrm{i} }^{\mathrm{c}}+ \mathrm{D}\right)}^{0.5}$, for the Venturi geometry used in this study and under the downstream valve sweep rate of 1/6 °/s, the sensitivity of the critical pressure ratio and the cavitation discharge coefficient with respect to the critical inlet pressure is valid within the Reynolds-number range [6.21 × 10⁴, 10.26 × 10⁴] covered by the experiments. Furthermore, because the critical inlet pressure is hysteresis-free within the practical equivalence margins specified in this study, the stated path independence/dependence of the cavitation inception conditions remains valid over the Reynolds-number range covered by the experiments.

Under a fixed downstream valve sweep rate that ensures a quasi-dynamic cavitation evolution and fixed fluid properties, the parameters A and B in the functional relationship between the critical pressure ratio and the critical inlet pressure, and the parameters C and D in the relationship for the cavitation discharge coefficient, are determined by the Venturi flow-passage geometry. Consequently, the characteristic pressure ratio and the characteristic cavitation discharge coefficient are dependent on the geometry. The hysteresis behavior of the cavitation inception conditions may also be strongly related to the geometry. Therefore, for convergent–divergent Venturis of similar design, whether the sensitivities of the inception conditions and the cavitation discharge coefficient resemble those reported here, and how the path independence and uniqueness of the inception conditions are linked to the geometry, can be investigated using the present dual-valve cavitation system.

5.4. Limitations, Contributions, and Prospects

Limited by a system sampling rate of 4 Hz, cavitation inception was determined using the intersection-based procedure in this study. During the transition from non-cavitating to cavitating flow, the instantaneous inception of cavitation and the stabilization of the flow rate do not occur simultaneously [15,17]. Therefore, the intersection-based inception provides an estimate of the instantaneous cavitation inception. In laboratory settings, the instantaneous inception can be identified in a transparent Venturi by high-speed imaging [17,27] or by detecting anomalies in acoustic or pressure signals at the onset of cavitation. In typical industrial applications, however, systems rarely include transparent Venturis, high-speed cameras, or high-sampling-rate acoustic or pressure sensors. Given these constraints, the intersection-based method proposed here, which requires only a very low sampling rate, provides a practical low-cost approach for determining the cavitation inception point.

Because the dual-valve cavitation system can only realize a family of shape-similar pressure-control paths in the non-cavitating stage, where the inlet pressure and the ambient pressure vary simultaneously, it is not potential to drive the flow to cavitation at the same critical inlet pressure along different paths. As a result, the boundary of the inception separation band remains difficult to determine experimentally. Moreover, since separated forward and backward inception conditions have been obtained only for the upstream valve opening of 75° in one set of quasi-dynamic tests, the starting point of the inception separation band, point N in Figure 11, has not yet been measured. Consequently, the cavitation inception point on an arbitrary pressure-control path cannot be predicted accurately from the inception locus together with the non-cavitating segment of that path alone.

Even so, the inception locus established here provides a foundation for inferring or comparing inception points across arbitrary control paths: even without direct inception testing, the intersection between a path’s non-cavitating segment and the inception locus offers insight into the approximate range of its inception point. In addition, the insensitivity of the critical pressure ratio to the critical inlet pressure beyond a threshold facilitates practical identification of inception along arbitrary paths. Furthermore, when a Venturi is used for stable-flow control via cavitation, the relationship between the cavitation discharge coefficient and the critical inlet pressure serves as a basis for choosing pressure-control paths to achieve the required steady flow rate.

With 512 Hz pressure transmitters now available, future experiments will use high-sampling-rate pressure records to determine the instantaneous cavitation inception point and compare it with the intersection-based estimate, thereby evaluating the accuracy of the inception criterion adopted here. A broader set of forward and backward quasi-dynamic measurements at upstream valve openings ≥ 35° will also be pursued to obtain multiple inception points and to infer the starting point of the inception separation band. In parallel, tests on Venturi tubes with different configurations will allow further examination of the relationships linking the critical pressure ratio to the critical inlet pressure and the cavitation discharge coefficient to the critical inlet pressure.

6. Conclusions

With a downstream valve sweep rate of 1/6 °/s, water temperature ranging from 23.7 to 25.4 °C and the dissolved oxygen concentration within 5.7 to 7.2 mg/L, the inlet pressure, ambient pressure and flow rate were measured during various quasi-dynamic cavitation evolution processes. Corresponding pressure- and flow-control paths were established. With the practical equivalence margins of ±3 × 10⁻³ MPa and ±3%, the sensitivity of the cavitation inception conditions and cavitation discharge coefficient was assessed using forward pressure- and flow-control paths acquired at a 4 Hz system sampling rate, while path independence and uniqueness of cavitation inception conditions were examined using the backward and forward pressure-control paths recorded at 512 Hz. The mean findings are as follows:

• As the critical inlet pressure approaches infinity, the critical pressure ratio tends to a constant characteristic pressure ratio ${\mathit{P}}_{\mathrm{r}}^{\mathrm{c}*} = 0.6364$ with a 95% t-confidence interval of [0.6301, 0.6380]. When the critical inlet pressure exceeds the threshold ${\mathit{P}}_{\mathrm{i}}^{\mathrm{c},\mathrm{thr}1}$ = 2.4496 MPa with a 95% t-confidence interval of [2.4172, 2.4814] MPa, the critical pressure ratio becomes effectively insensitive to critical inlet pressure.
• As the critical inlet pressure approaches infinity, the critical discharge coefficient approaches a characteristic value $C_{\mathrm{d}}^{\mathrm{c}*} = 0.5573$ MPa with a 95% t-confidence interval of [0.5563, 0.5582]. When the critical inlet pressure exceeds the threshold ${\mathit{P}}_{\mathrm{i}}^{\mathrm{c},\mathrm{thr}2}$ = 1.7167 MPa with a 95% t-confidence interval of [1.6780, 1.7570] MPa, the cavitation discharge coefficient becomes effectively insensitive to critical inlet pressure.
• For the pressure-control paths generated by the dual-valve cavitation system and Venturi tube in this study, cavitation inception conditions were found to be unique.
• Hysteresis in the critical pressure ratio appears at a critical inlet pressure ranging from 0.40 to 0.45 MPa. For critical inlet pressures greater than 0.45 MPa, path independence of the cavitation inception conditions no longer holds.

The findings and established inception locus provide a path independent reference for inferring inception across control paths, while the demonstrated invariance of critical pressure ratio and cavitation discharge coefficient beyond identified thresholds provides actionable rules for inception and stabilized flow rate control.